Möbius voting for surface correspondence

TLDR

Isometries form a subset of the Möbius group, which has only six degrees of freedom for spheres and three for discs. The study aims to develop an efficient, automatic algorithm for discovering point correspondences between surfaces that are approximately or partially isometric. The algorithm, called Möbius Voting, iteratively samples triplets of points, computes closed‑form Möbius transformations via mid‑edge flattening to the complex plane, maps the point sets into a canonical frame, and votes for correspondences weighted by a transportation‑type distance that models deviations from isometry. The method yields a fuzzy correspondence matrix that is converted to a permutation matrix, producing discrete correspondences with confidence values, and experiments show it can automatically find dozens of correspondences across different object types and poses even under extreme deformation.

Abstract

The goal of our work is to develop an efficient, automatic algorithm for discovering point correspondences between surfaces that are approximately and/or partially isometric. Our approach is based on three observations. First, isometries are a subset of the Möbius group, which has low-dimensionality -- six degrees of freedom for topological spheres, and three for topological discs. Second, computing the Möbius transformation that interpolates any three points can be computed in closed-form after a mid-edge flattening to the complex plane. Third, deviations from isometry can be modeled by a transportation-type distance between corresponding points in that plane. Motivated by these observations, we have developed a Möbius Voting algorithm that iteratively: 1) samples a triplet of three random points from each of two point sets, 2) uses the Möbius transformations defined by those triplets to map both point sets into a canonical coordinate frame on the complex plane, and 3) produces "votes" for predicted correspondences between the mutually closest points with magnitude representing their estimated deviation from isometry. The result of this process is a fuzzy correspondence matrix, which is converted to a permutation matrix with simple matrix operations and output as a discrete set of point correspondences with confidence values. The main advantage of this algorithm is that it can find intrinsic point correspondences in cases of extreme deformation. During experiments with a variety of data sets, we find that it is able to find dozens of point correspondences between different object types in different poses fully automatically.